Integrand size = 38, antiderivative size = 214 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (A-i B) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{a^{3/2} d}+\frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d} \]
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Time = 0.83 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4326, 3677, 3679, 12, 3625, 211} \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}} \]
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Rule 12
Rule 211
Rule 3625
Rule 3677
Rule 3679
Rule 4326
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx \\ & = \frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {1}{2} a (7 A+i B)-2 a (i A-B) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2} \\ & = \frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)} \left (\frac {1}{4} a^2 (25 A+7 i B)-\frac {1}{2} a^2 (11 i A-5 B) \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{3 a^4} \\ & = \frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {3 a^3 (i A+B) \sqrt {a+i a \tan (c+d x)}}{8 \sqrt {\tan (c+d x)}} \, dx}{3 a^5} \\ & = \frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}+\frac {\left ((i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx}{4 a^2} \\ & = \frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d}-\frac {\left (i (i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{-i a-2 a^2 x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{2 d} \\ & = \frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (i A+B) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{a^{3/2} d}+\frac {(A+i B) \sqrt {\cot (c+d x)}}{3 d (a+i a \tan (c+d x))^{3/2}}+\frac {(11 A+5 i B) \sqrt {\cot (c+d x)}}{6 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{6 a^2 d} \\ \end{align*}
Time = 7.15 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.92 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {i \left (3 \sqrt {2} (A-i B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) (i+\cot (c+d x)) \sqrt {a+i a \tan (c+d x)}+\frac {2 a (-39 i A+9 B-12 A \cot (c+d x)+(25 A+7 i B) \tan (c+d x))}{\sqrt {i a \tan (c+d x)}}\right )}{12 d \cot ^{\frac {5}{2}}(c+d x) (i a \tan (c+d x))^{3/2} (-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (172 ) = 344\).
Time = 0.61 (sec) , antiderivative size = 939, normalized size of antiderivative = 4.39
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(939\) |
default | \(\text {Expression too large to display}\) | \(939\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (162) = 324\).
Time = 0.26 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.17 \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} + {\left (A - i \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 3 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (3 i \, d x + 3 i \, c\right )} \log \left (-\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} + {\left (A - i \, B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) + \sqrt {2} {\left (2 \, {\left (19 \, A + 4 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (13 \, A + 7 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - A - i \, B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{12 \, a^{2} d} \]
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\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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